Problem 58 - Algebra 2 and Trigonometry

Question

In \(51-58,\) find the smallest positive value of \(\theta\) to the nearest minute. $$ \tan \theta=0.0892 $$

Step-by-Step Solution

Answer

The smallest positive value of \( \theta \) is 5 degrees and 5 minutes.

Step 1: Understand the Tangent Function

The tangent function, \( \tan \theta \), relates the angle \( \theta \) in a right-angled triangle to the ratio of the opposite side over the adjacent side. We need to find the angle \( \theta \) such that \( \tan \theta = 0.0892 \).

Step 2: Use Inverse Tangent

To find the angle \( \theta \), we use the inverse tangent function, \( \theta = \tan^{-1}(0.0892) \). This operation will give us \( \theta \) in degrees.

Step 3: Calculate using a Calculator

Using a calculator, find the value of \( \theta = \tan^{-1}(0.0892) \). This should give \( \theta \approx 5.091 \) degrees.

Step 4: Convert Decimal Degrees to Minutes

The decimal degree \( 5.091 \) needs to be converted into degrees and minutes. The whole number part is the degrees (5 degrees), and we convert the decimal part into minutes. Multiply 0.091 by 60 to find the minutes. \( 0.091 \times 60 = 5.46 \).

Step 5: Round Minutes to Nearest Whole Number

We round 5.46 minutes to the nearest whole number which is 5 minutes.

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions, along with sine and cosine. It is crucial in understanding the relationships between the angles and sides of a right-angled triangle. For an angle \( \theta \), the tangent function, written as \( \tan \theta \), is defined as the ratio of the length of the opposite side to the length of the adjacent side.

• Definition: In mathematical terms, \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \).
• Uses: Tangent is useful in various applications such as calculating heights of tall objects, navigation, and solving problems involving angular measurements.

When you are given \( \tan \theta = 0.0892 \), it means the ratio of the opposite to the adjacent side for this angle \( \theta \) is 0.0892. To find the exact angle \( \theta \) from this ratio, use the inverse tangent function.

Angle Calculation

To calculate the specific angle \( \theta \) when you know its tangent, you use the inverse tangent function, also called \( \arctan \) or \( \tan^{-1} \). This function takes a ratio as input and returns an angle.

• Inverse Tangent: Denoted as \( \theta = \tan^{-1}(x) \), where \( x \) is a given ratio.
• Example Calculation: For \( \tan \theta = 0.0892 \), you calculate \( \theta = \tan^{-1}(0.0892) \).

Using a scientific calculator will help you perform this calculation quickly. Typically, the result for \( \theta \) is in degrees. For our problem, \( \theta \) is approximately 5.091 degrees.
This precise angle is vital in various any geometry or navigation tasks, offering accuracy in measurements.

Degrees to Minutes Conversion

Often, angles are not left as decimal degrees because they need higher precision, especially in navigation or astronomy. To address this, we convert decimal degrees into degrees and minutes.

• Understanding Degrees and Minutes: One degree is equal to 60 minutes (1° = 60'). This allows us to break down a decimal degree into more precise parts.
• Conversion Process: Only the decimal part of the degree is converted to minutes. Multiply this decimal by 60 to change it to minutes.

For \( \theta \approx 5.091 \), we have 5 degrees and \( 0.091 \times 60 = 5.46 \) minutes. Since 5.46 is not a whole number, round it to the nearest minute, giving you 5 degrees and 5 minutes as the angle \( \theta \).
This conversion is necessary for more detailed calculations without losing precision due to rounding off decimal places.